If \({\mathbf {x}}\)  and the noise are independent Gaussian random variables then Eq.  REF is the covariance of the posterior distribution for \({\mathbf {g}}({\mathbf {x}})\)  given \({\mathbf {m}}_{{S}}({\mathbf {x}})\) .
A low-dimensional representation of the state and its covariance are usually found from data via principal component analysis (PCA) [1]} or proper orthogonal decomposition (POD) [2]} when an analytical model is not available.
